# [seqfan] Re: Fwd: A nice (decimal) property of 78

f.firoozbakht at sci.ui.ac.ir f.firoozbakht at sci.ui.ac.ir
Thu Nov 6 21:27:47 CET 2008

```> n = a concat b, phi(n) = phi(a) * phi(b)

I think your sequence (78, 780, 897, 918, 1179, 1365, 1776, 2574, 2598,
2967, 3168, 3762, 4758, 5775, 5796, ...) is a nice sequence and it should
be submitted. Can you please do it?

I proved an interesting property of this sequence: " If n is in the
sequence then 10^m*n for each natural number m is also in the sequence. "
The proof is easy and I did it in four cases.
So all numbers of the form 10^m*78, 10^m*897, 10^m*918, 10^m*1179, 10^m*1365,
... are in the sequence.

Thanks,
Farideh

Quoting David Wilson <dwilson at gambitcomm.com>:

> The n with
>
> n = a concat b, phi(n) = phi(a) * phi(b)
>
> seem to be fairly common, e.g:
>
> phi(78) = phi(7)*phi(8)
> phi(780) = phi(7)*phi(80)
> phi(897) = phi(89)*phi(7)
> phi(918) = phi(91)*phi(8)
> phi(1179) = phi(11)*phi(79)
>
> The list starts
>
> 78 780 897 918 1179 1365 1776 2574 2598 2967 3168 3762 4758 5775 5796
> 7800 7875 7917 8217 8970 9180 9576 11790 13650 13662 13875 13896 14391
> 17760 18564 18858 19812 20097 25740 25935 25974 25980 27573 28776 28779
>
> Maximilian Hasler wrote:
>> (10:05) gp >
>> for(i=11,9999,i%10|next;eulerphi(i)==eulerphi(i\10)*eulerphi(i%10)&print1(i","))
>> 78,897,918,2598,4758,7917,8217,
>>
>> Maximilian
>> PS: can one have eulerphi(i)=product(eulerphi( k-th digit of i )) for i>78 ?
>> I don't think so; it seems as if there would be strict ">" for all i
>> different from 78.
>>
>>
>
>
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>
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>

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